We start considering the system of integral equations
\(\varphi (x,t)= e^{it \partial _x^2}\varphi _0(x)+i\lambda \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi })(x,t^{\prime })\, dt^{\prime },\) 
\(\psi (x,t)=e^{it \partial _x^2}\psi _0(x)+i\lambda \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(-\psi ^2\overline{\varphi })(x,t^{\prime })\, dt^{\prime },\) 
which can be combined using vector notation as
\((\varphi ,\psi )=\Phi (\varphi ,\psi ),\) 
for
\(\Phi (\varphi ,\psi ):= e^{it \partial _x^2}(\varphi _0,\psi _0)(x)+i\lambda \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi },-\psi ^2\overline{\varphi })(x,t^{\prime })\, dt^{\prime }. \) 
Notice that, by using Duhamel's principle, any solution of the differential equations (REF ) is also a solution of the integral equations (REF ). The converse is not true in general. However, for initial data \(\varphi _0,\psi _0 \in H^2(\mathbb {R})\)  one has that a solution of (REF ) also satisfies the NLS system (REF ) (proceed as in [1]}).
Therefore, to prove this Proposition we are going to show the existence and uniqueness of solutions of (REF ) verifying the persistency property (REF ).
For some constants \(a,T>0\) , that will be conveniently fixed later, consider the space
\(X^a_T :=\lbrace \varphi \text{ : }\Vert \varphi \Vert _{X_T} \le a\rbrace ,\) 
endowed with the norm
\(\Vert  \varphi \Vert _{X_T}:=\Vert  \varphi \Vert _{Y_T}+\Vert |x|^{r} \varphi \Vert _{L^\infty _T L^2_x},\) 
where
\(\Vert  \varphi \Vert _{Y_T}:={\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial ^2_x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\) 
with \({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \cdot \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\)  introduced in ().
Moreover, we define
\(\Vert (\varphi ,\psi )\Vert _{X_T \times X_T}:=\Vert \varphi \Vert _{X_T}+\Vert \psi \Vert _{X_T},\) 
and say that \((\varphi ,\psi ) \in X^a_T\times X^a_T\)  provided that
\(\Vert (\varphi ,\psi )\Vert _{X_T \times X_T}\le 2a.\) 
By the Banach fixed-point theorem, it is enough to establish that the mapping \(\Phi \)  given in (REF ) is a contraction on the complete metric space \(X^a_T \times X^a_T\) .
Step 1: \(\Phi \)  is well defined.
Our first goal is to see that \(\Phi : X^a_T \times X^a_T \longmapsto X^a_T \times X^a_T\) , that is, for \(\varphi , \psi \in X^a_T\)  we need to show that
\(\Vert \Phi (\varphi , \psi )\Vert _{X_T \times X_T}\le 2a.\) 
   
Recalling well-known results for the unweighted case \(m=0\)  (see for example
[2]} and the references therein) one can deduce that
\(\Vert \Phi (\varphi , \psi )\Vert _{Y_{T} \times Y_{T}}\lesssim \Vert \varphi _0\Vert _{H^2}+\Vert \psi _0\Vert _{H^2}+T^{\theta }(\Vert  \varphi \Vert _{Y_T}^2 \,\Vert  \psi \Vert _{Y_T}+\Vert  \varphi \Vert _{Y_T} \,\Vert  \psi \Vert _{Y_T}^2),\) 
for certain \(\theta >0\) .
According to
(REF ) and (REF ), it remains to control the sum of the terms

2
\(& I_1 :=\Vert |x|^{r}e^{it \partial _x^2}(\varphi _0)\Vert _{L^\infty _T L^2_x}, \\& I_2 :=\Vert |x|^{r}e^{it \partial _x^2}(\psi _0)\Vert _{L^\infty _T L^2_x},\) 
\(& I_3 :=\Big \Vert |x|^{r}\int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi })(x,t^{\prime })\, dt^{\prime }\Big \Vert _{L^\infty _T L^2_x}, \\& I_4 :=\Big \Vert |x|^{r}\int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\psi ^2\overline{\varphi })(x,t^{\prime })\, dt^{\prime }\Big \Vert _{L^\infty _T L^2_x} .\) 


By symmetry, it is enough to consider \(I_1\)  and \(I_3\) .
For the analysis of \(I_1\) , we simply use
(REF ), and (REF ) to get
\(I_1& \lesssim T^{r/2} \Vert  \varphi _0 \Vert _{L^2}+T^{r} \Vert  D^r_x \varphi _0 \Vert _{L^2}+\Vert |x|^r \varphi _0 \Vert _{L^2}\nonumber \\& \lesssim (T^{r/2}+T^{r})\Vert \varphi _0\Vert _{H^{1}}+\Vert |x|^{r}\varphi _0\Vert _{L^2}.\) 
As for \(I_{3}\) , once again (REF ) produces
\(I_{3}& \le \int _{0}^T\Big \Vert |x|^{r}e^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi })(x,t^{\prime })\Big \Vert _{L^\infty _T L^2_x}\, dt^{\prime } \nonumber \\& \lesssim T^{1+r/2} \,\Vert \varphi ^2 \psi \Vert _{L^\infty _T L^2_x}+T^{1+r} \,\Vert D_x^{r}(\varphi ^2\overline{\psi })\Vert _{L^\infty _T L^2_x}+T \, \Vert  |x|^{r} \varphi ^2 \psi \Vert _{L^\infty _T L^2_{x}} \nonumber \\& \lesssim (T^{1+r/2} + T^{1+r}) \,\Vert \varphi ^2 \overline{\psi }\Vert _{L^\infty _TH^1_{x}}+T \,\Vert \varphi \Vert _{L^\infty _T L_x^{\infty }} \,\Vert  \psi \Vert _{L^\infty _T L_x^{\infty }} \,\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2}\nonumber \\& \lesssim (T^{1+r/2} + T^{1+r}) \,\,\Vert \varphi \Vert _{L^\infty _TH^1_{x}}^2\Vert \psi \Vert _{L^\infty _TH^1_{x}}+T \,\Vert \varphi \Vert _{L^\infty _T H_x^{1}} \,\Vert  \psi \Vert _{L^\infty _T H_x^{1}} \,\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2} \nonumber \\& \lesssim (T^{1+r/2} + T^{1+r} + T)\Vert \varphi \Vert _{X_T}^2 \,\Vert  \psi \Vert _{X_T}\) 

where we also used the Sobolev properties
(REF )
and
(REF ).
Combining
(REF ),
(REF ) and
(REF )
(and taking into consideration the analogous estimates for \(I_2\)  and \(I_4\) ) we deduce that,
for every \(\varphi , \psi \in X^a_T\) ,
\(\Vert \Phi (\varphi , \psi )\Vert _{X_T \times X_T}& \le C\Big ((1+T^{\theta })(\Vert \varphi _0\Vert _{H^{2}}+\Vert \psi _0\Vert _{H^{2}})+\Vert |x|^{r}\varphi _0\Vert _{L^2}+\Vert |x|^{r}\psi _0\Vert _{L^2}\Big ) \\& \quad +C \,T^{\theta } \, a^3,\) 

for some \(C, \theta >0\) .
Finally, set
\(a:=C(\Vert \varphi _0\Vert _{H^{2}}+\Vert \psi _0\Vert _{H^{2}}+\Vert |x|^{r}\varphi _0\Vert _{L^2}+\Vert |x|^{r}\psi _0\Vert _{L^2})\) 
and chose \(T\)  small enough verifying
\(C \, T^{\theta } \,(\Vert \varphi _0\Vert _{H^{2}}+\Vert \psi _0\Vert _{H^{2}})+C \, T^{\theta } \,a^3\le a,\) 
to conclude (REF ).
Step 2: \(\Phi \)  is a contraction.
Now the task is to show the existence of a constant \(0<K<1\) 
verifying that
\(\Vert \Phi (\varphi , \psi )- \Phi (\widetilde{\varphi }, \widetilde{\psi })\Vert _{X_T \times X_T}\le K \Vert (\varphi , \psi )-(\widetilde{\varphi }, \widetilde{\psi })\Vert _{X_T\times X_T},\) 
for \(\varphi , \widetilde{\varphi },\psi , \widetilde{\psi } \in X^a_T\) .
It is known that (see
[2]}),
for some \(\theta >0\) ,
\(\Vert \Phi (\varphi , \psi )- \Phi (\widetilde{\varphi }, \widetilde{\psi })\Vert _{Y_T \times Y_T}& \lesssim T^\theta \big (\Vert \varphi \Vert _{Y_T}+\Vert \widetilde{\varphi }\Vert _{Y_T}+\Vert \psi \Vert _{Y_T}+\Vert \widetilde{\psi }\Vert _{Y_T}\big )^2 \nonumber \\& \quad \times \Vert (\varphi , \psi )-(\widetilde{\varphi }, \widetilde{\psi })\Vert _{Y_T \times Y_T}.\) 

Hence, we concentrate on the terms
\(J_1&:=\Big \Vert  |x|^{r}\int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2\overline{\psi }-\widetilde{\varphi }^2\overline{\widetilde{\psi }})(x,t^{\prime })\, dt^{\prime }\Big \Vert _{L^\infty _T L^2_x}, \\J_2&:=\Big \Vert  |x|^{r}\int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\psi ^2\overline{\varphi }-\widetilde{\psi }^2\overline{\widetilde{\varphi }})(x,t^{\prime })\, dt^{\prime }\Big \Vert _{L^\infty _T L^2_x}.\) 

We only analyze \(J_1\) , \(J_2\)  can be treated analogously.
Next, observe that we can rewrite the difference of nonlinear factors as follows
\(\varphi ^2\overline{\psi }-\widetilde{\varphi }^2\overline{\widetilde{\psi }}=\varphi ^2(\overline{\psi }-\overline{\widetilde{\psi }})+\overline{\widetilde{\psi }}(\varphi +\widetilde{\varphi })(\varphi -\widetilde{\varphi }).\) 
   
Then, proceeding as in
(REF )
and
(REF )
above, we get
\(J_1& \lesssim T \,\Vert  |x|^{r}(\varphi ^2\overline{\psi }-\widetilde{\varphi }^2\overline{\widetilde{\psi }}) \Vert _{L^\infty _T L^2_{x}} +(T^{1+r/2}+T^{1+r}) \,\Vert  \varphi ^2\overline{\psi }-\widetilde{\varphi }^2\overline{\widetilde{\psi }} \Vert _{L^\infty _T H^2_x} \nonumber \\& \le T \,\Big (\Vert  |x|^{r} \varphi \Vert _{L^\infty _T L^2_{x}} \,\Vert  \varphi \Vert _{L^\infty _T H^2_{x}} \,\Vert  \psi -\widetilde{\psi } \Vert _{L^\infty _T H^2_{x}} \nonumber \\& \qquad +\Vert  |x|^{r} \widetilde{\psi }\Vert _{L^\infty _T L^2_{x}} \,\Vert  \varphi \Vert _{L^\infty _T H^2_{x}} \,\Vert  \varphi -\widetilde{\varphi }\Vert _{L^\infty _T H^2_{x}} \nonumber \\& \qquad +\Vert  |x|^{r} \widetilde{\psi }\Vert _{L^\infty _T L^2_{x}} \,\Vert  \widetilde{\varphi } \Vert _{L^\infty _T H^2_{x}} \,\Vert  \varphi -\widetilde{\varphi }\Vert _{L^\infty _T H^2_{x}}\Big ) \nonumber \\& \quad +(T^{1+r/2}+T^{1+r})\Big (\Vert \varphi \Vert _{L^\infty _T H^2_{x}}^2 \,\Vert  \psi -\widetilde{\psi } \Vert _{L^\infty _T H^2_{x}}+\Vert \widetilde{\psi }\Vert _{L^\infty _T H^2_{x}} \,\Vert \varphi \Vert _{L^\infty _T H^2_{x}} \,\Vert  \varphi -\widetilde{\varphi } \Vert _{L^\infty _T H^2_{x}} \nonumber \\& \hspace{113.81102pt} +\Vert \widetilde{\psi }\Vert _{L^\infty _T H^2_{x}} \,\Vert \widetilde{\varphi }\Vert _{L^\infty _T H^2_{x}} \,\Vert  \varphi -\widetilde{\varphi } \Vert _{L^\infty _T H^2_{x}}\Big ).\) 
   
Putting together (REF ),
(REF ) and the corresponding symmetric estimate for \(J_2\)  we get, for \(\varphi , \widetilde{\varphi },\psi , \widetilde{\psi } \in X^a_T\) ,
\(\Vert \Phi (\varphi , \psi )- \Phi (\widetilde{\varphi }, \widetilde{\psi })\Vert _{X_T \times X_T}& \le C T^\theta a^2\Vert (\varphi , \psi )-(\widetilde{\varphi }, \widetilde{\psi })\Vert _{X_T\times X_T},\) 

for certain \(C,\theta >0\) .
Therefore, it is possible to pick \(T\)  satisfying simultaneously (REF ) and \(C T^\theta a^2 <1\) .
